The way I have presented it is showing how mathematicians think. Get an idea, try it out, if it appears to work then attempt to produce a logical and mathematically sound derivation.
(This last part I have not included)
The idea is that wherever you have operations on things, and one operation can be followed by another of the same type, then you can consider the combinations of the operations separately from the things being operated on. The result is a new type of algebra, in this case the algebra of rotations.
Read on . . .

angle 90 deg or pi/2
Y = x, and X = -y
so (1,0) goes to (0,1) and (-1,0) goes to (0,-1)
and (0,1) goes to (-1,0) and (0,-1) goes to (1,0)
Let us call this transformation Q (for a quarter turn)

Then H(x,y) = (-x,-y)
and Q(x,y) = (-y,x)

Applying H twice we have H(H(x,y)) = (x,y) and if we are bold we can write HH(x,y) = (x,y)
and then HH = I, where I is the identity or do nothing transformation.
In the same way we find QQ = H

Now I is like multiplying the coodinates by 1
and H is like multiplying the coordinates by -1
This is not too outrageous, as a dilation can be seen as a multiplication of the coordinates by a number <> 1

So what is Q ?
Let us suppose that it is some sort of a number, definitely not a normal one,
and let its value be called k.
What we can be fairly sure of is that k does not multiply each of the coordinates.
This appears to be meaningful only for the normal numbers.

Now the “number” k describes a rotation of 90, so we would expect that the square root of k to describe a rotation of 45

At this point it helps if the reader is familiar with extending the rational numbers by the introduction of the square root of 2 (a surd, although this jargon seems to have disappeared).

Let us assume that sqrt(k) is a simple combination of a normal number and a multiple of k:
sqrt(k) = a + bk
Then k = sqr(a) + sqr(b)*sqr(k) + 2abk, and sqr(k) = -1
which gives k = sqr(a)-sqr(b) + 2abk and then (2ab-1)k = sqr(a) – sqr(b)

From this, since k is not a normal number, 2ab = 1 and sqr(a) = sqr(b)
which gives a = b and then a = b = 1/root(2)

Now we have a “number” representing a 45 degree rotation. namely
(1/root(2)*(1 + k)

If we plot this and the other rotation numbers as points on a coordinate axis grid with ordinary numbers horizontally and k numbers vertically we see that all the points are on the unit circle, at positions corresponding to the rotation angles they describe.

OMG there must be something in this ! ! !

The continuation is left to the reader (as in some Victorian novels)

ps. root() and sqrt() are square root functions, and sqr() is the squaring function .

3 responses to “Complex Numbers via Rigid Motions”

When I think about such things and read rigid motion, I am unable to get the exponential function out of my head. I still remember marvelling at the beauty of learning about infinitesimal vectors for small rotations … then using Taylor’s expansion to represent larger rotations … and then projecting the exponential function to the 2D complex plan – and then anything fell into place.
I really wonder when is the earliest possible time to teach that? Or at least using the exponential function instead of cosine and sine? Explaining Taylor’s expansion is not that complex once you understand how to differentiate polynomials… and then all is the relation between cosine, sine, and the exponential function – to be derived from their expansions. And all those rules for cos(a + b) etc. can be derived easily.

Thinking about this I am reminded about the discussion on so-called Geometrical Algebra: http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf . The autor challenges the traditional way of introducing math in physics courses and suggests to dealing with vector calculus in a completely different way from the start, instead of learning seemlingly separate laws (Gauss, Stokes…) only to learn years later – or never – that there would have been a way of unification… which does not sound too complicated after all if you imagine this geometrial algebra would be the first thing you learn?
So in related way, I am not sure when is the right time to introduce calculus, or the relation between trig functions and the exponential function?

I just read the Oersted lecture up to the point of the definition and explanation of the Geometrical Product – this guy is a genius – who would have thought that the dot product and the cross product could be related in such a simple and elegant way. I am going to read the rest in smaller bites.
The teaching of calculus here in the USA is pathetic. Starts with the idea and immediate use of limits of functions as h—>0 without a proper grounding in limits and continuity, and not a serious look at limits of sequences and series.
Then it’s off to do the sums. Differentiation of you name it, integration of you’ve probably forgotten the name of it, some thing weirdly named u-substitution (I guess it doesn’t work if you used t or v), no numerical integration methods, and no sign of any use of calculus for solving equations.
In other words, all the stuff for tests and very little else. And precious little on real applications. Don’t even ask about differential equations.
A stroy from my teaching of control and electrical engineering students: I happened to mention that an integrator did integration of a signal in the calculus sense. “What? Really?” was the general response. I was shocked !

I think that the ideas of varying rates of change can and should be introduced gradually, and graphically. The actual rate of change function (the difficult idea) can be found without the use of limits for most common functions using coordinate geometry methods. See my posts on Calculus Without Limits.

I like the complex numbers by rotations intro, it was years before I was happy with the argument of the product of z1 and z2 = the sum of the arguments of each of z1 and z2. It didn’t make any sense.

One of the horrors of school math is defining e as the limit of that mess with powers of 1/n, and then baldly stating the power series expansion, and then the power series for sin and cos, and then justifying the formulae for their derivatives by term by term differentiation.